ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 4 Solutions [Numerical Methods]

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1 ENGI 3 Advaced Calculus or Egieerig Facult o Egieerig ad Applied Sciece Problem Set Solutios [Numerical Methods]. Use Simpso s rule with our itervals to estimate I si d a, b, h a si si.889 si 3 si b si h I Thereore, correct to d.p., si d.8 A Ecel spreadsheet ile is available or this questio. [Maple reports that the eact value is Oe ca see that this approimatio is quite rough, with a error o over %. The error rom Simpso s rule alls uder % or.]

2 ENGI 3 Problem Set Solutios Page o. Use the trapezoidal rule with si itervals to estimate I d. Use Simpso s rule with si itervals to estimate I. Fid the eact value o I ad commet o the accurac o the two approimatios. a, b, h Trapezoidal rule: h I Simpso s rule: h.373 I Correct to three decimal places, these schemes produce the approimatios d.78 (Trapezoidal) ad d.785 (Simpso s) Eact: Either quote a stadard itegral or use the substitutio ta sec d sec ta d d I d I Arcta d The trapezoidal rule is o b less tha.5%, while Simpso s rule is good to at least ive sigiicat igures. A Ecel ile is available or this problem.

3 ENGI 3 Problem Set Solutios Page 3 o 3. Use Simpso s rule with eight itervals to estimate I e d. t [The error uctio is deied to be er e dt. It has multiple uses, icludig the calculatio o probabilities or radom quatities that ollow a ormal distributio. It caot be evaluated eactl, ecept or a ew special choices o.] a, b, 8 h.5 8 e.5 e e e e e e e.53 8 e Simpso s rule: h I Thereore, correct to 3 d.p., e d.77 [For this itegrad, Simpso s rule provides a ver accurate approimatio. The eact value is er.78. Eve with ol our itervals, the error is well uder.%.] A Ecel ile is available or this problem.

4 ENGI 3 Problem Set Solutios Page o. I Problem Set Questio 3, the total arc legth alog a oe loop o the curve r / cos3 was oud to be L 9 sec3 5cos3 d. Eplai wh a simple trapezoidal or Simpso s rule ca ot be used to estimate this arc legth. The itegrad icludes sec3. lim sec3. / The umerical approimatio thereore diverges to iiit. 5. Use Newto s method (ad coirm with a graphical method) to id the value o tah e e e to si sigiicat igures, where tah. e e e We are required to solve tah tah e tah e d e e e e e d e e (quotiet rule o dieretiatio) e e e e e e e e sech = tah is the logistic curve, with rage < < +: A reasoable irst guess or the clearl uique solutio to tah = / is

ENGI 3 Problem Set Solutios Page 5 o 5 (cotiued) At the core o Newto s method is the iteratio Completig a table o values:. e e e e.5.37 88....78 7....8 9....58 9.... 85....75 85.... 35....59 35

5 ENGI 3 Problem Set Solutios Page 5 o 5 (cotiued) At the core o Newto s method is the iteratio Completig a table o values:. e e e e A Ecel ile coirms these calculatios. Ater roud-o, ad all subsequet are idetical to si sigiicat igures. Thereore, correct to si sigiicat igures, tah.593 A series o zooms o the graph = tah coirms this aswer:

6 ENGI 3 Problem Set Solutios Page o 5 (cotiued) which is cosistet with tah.593. Additioal Note: tah is the hperbolic taget o. Oe o the was i which the hperbolic uctios are related to the more amiliar trigoometric uctios is through the epoetial uctio ad j : j j j j e e e e e e e e si, sih, cos, cosh j. As see i Questio 5, whe Newto s method works well, the covergece ca be quite rapid. Now cosider the solutios to cos.3. As see rom the plot below, there are ive solutios to this equatio. Let us tr to id the largest root.

7 ENGI 3 Problem Set Solutios Page 7 o (a) Use Newto s method with a iitial guess o 7 to id a solutio correct to ive decimal places. Deie.3 cos.3 si () '() Thereore, correct to ive decimal places, which is ideed the largest root..53 (b) Use Newto s method with a iitial guess o to id a solutio to 5 d.p. () '() Thereore, correct to ive decimal places, which is the secod-largest root, ot the largest root.

8 ENGI 3 Problem Set Solutios Page 8 o (c) Use Newto s method with a iitial guess o. to id a solutio to 5 d.p. () '() Thereore, correct to ive decimal places,.3 which is the middle root, ot the largest root. (d) Use Newto s method with a iitial guess o.999 to id a solutio to 5 d.p. () '() Thereore, correct to ive decimal places, which is ideed the largest root..53

9 ENGI 3 Problem Set Solutios Page 9 o (e) Commet briel o these dieret results. What is the sigle greatest actor i the variabilit o these results? The two largest roots are closel spaced. Thereore a iitial guess that is ot close eough to the largest root (as i part (b)) ca coverge o the [earer] secod largest root istead. A Ecel ile is available or this solutio. Also is close to zero ear the two largest roots ad Newto s method icludes a divisio b. This ca result i wild istabilities i Newto s method. I parts (c) ad (d), iitial guesses ol. apart coverged o ver dieret solutios. Newto s method does t coverge at all or a iitial guess o.5! [Tr it.]

10 ENGI 3 Problem Set Solutios Page o 7. This questio provides some practice i algebra ad reers to page 5. o the e e lecture otes. From the deiitio sih, veri that sih l l e e Let sih sih e e e e e e e But e. Thereore discard the egative root. Also e sih l l l l Thereore sih l l Back to the ide o solutios

Newton s Method. y f x 1 x x 1 f x 1. Letting y 0 and solving for x produces. x x 1 f x 1. x 1. x 2 x 1 f x 1. f x 1. x 3 x 2 f x 2 f x 2.

Newton s Method. y f x 1 x x 1 f x 1. Letting y 0 and solving for x produces. x x 1 f x 1. x 1. x 2 x 1 f x 1. f x 1. x 3 x 2 f x 2 f x 2. 460_008.qd //04 :7 PM Page 9 SECTION.8 Newto s Method 9 (a) a a Sectio.8 (, ( )) (, ( )) Taget lie c Taget lie c b (b) The -itercept o the taget lie approimates the zero o. Figure.60 b Newto s Method Approimate